The IBVP is related to many important aspects of general relativity and the Einstein equations, such as numerical relativity, the construction of asymptotically Anti-de-Sitter spacetimes, timelike hypersurfaces emerging as the boundaries of the support of massive matter fields or the study of gravitational waves in a cavity [1]} and their nonlinear interactions. This problem was first addressed for the Einstein equations in the seminal work of Friedrich-Nagy [2]}, as well as by Friedrich [3]} in the related Anti-de-Sitter settingSee also [4]}, [5]} for extensions and other proofs of well-posedness in the Anti-de-Sitter case.. Well-posedness of the IBVP has since been obtained in generalized wave coordinates, see [6]} or the recent [7]}To be more precise, the boundary data in [7]} relies on an auxiliary wave map equation akin to generalized wave coordinates. This introduces a geometric framework to address the IBVP, albeit for the Einstein equations coupled to the auxiliary wave map equation. and for various first and second order systems derived from the ADM formulation of the Einstein equations, see for instance [9]}, [10]} and previous work in numerics [11]}, [12]}. We refer the reader to [13]} for an extensive review of the subject.
